Question

Suppose a and b are rational roots, and α,βbe the roots of $x^2$+2ax+b=0, then the equation with rational coefficients one of whose roots is α+β+ $\sqrt[2]{2{\alpha }^2+{\beta }^2}$ is

• A. x²+ 4ax +2b=0
• B. x²+ 4ax -2b=0
• C. x^{2 _} 4ax +2b=0
• D. x^2_ 4ax -2b=0

Answer 1

The correct option is A

x²+ 4ax +2b=0

We have , α+β= -2a , αβ= b

Let y= α+β+ $\sqrt[2]{2{\alpha }^2+{\beta }^2}$

=> y = -2a +$\sqrt[2]{2(\alpha +\beta {)}^2}-2\alpha \beta$

=> ${(y+2a)}^2$ = ${(-2a)}^2$ -2b

=>$y^2$+ 4$a^2$+4ay= 4$a^2$-2b

=> $y^2$+ 4ay+2b =0

Changing y to x , we have $x^2$+ 4ax +2b=0

This is the required equation .